Navigation instrument



June 25, 1935. v B BUCK 2,006,127

Y NAVIGATION INSTRUMENT Filed Jan. 16, 1955 2 Sheets-Sheet 1 Jue 25, 1935. vv. B. BUCK 2,006,127

NAVIGATION INSTRUMENT Filed Jan. 16, 1955 -2 Sheets-Sheet 2 me/who@ I W/Cof Uc/,

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Patented June 25, 1935 Victor B.l Buck, NewYork, N. Y. y Application January 16, 1935, Serial No. 2,103 Y 10 claims.'

vMy invention relates to an apparatus for solving graphically navigation problems without the use of tables, and more particularly to an arrangement for solving the general formula for 5 spherical triangles Aknown as the law of -cosines.

The general formula used in navigation and in other problems of spherical trigonometry is as follows:

lo cos arr=cos b cos c-l-sin b sin c cos A (a) In navigation problems, the conventional nomenclature instituted by Bowditch makes this formula read as follows:

sin Mor cos e) :sin L sin d-i-cos Ly cos d cos t (b)I In this formula (b), L represents the latitude, h `the altitude, d. the declination, and t the hour In order to explain the theory of the present device, reference is' made to Fig. 1 of the drawings. The vertical line AB shown in the drawings is made equal in length to the diameter, or to two radii, of a circle of unit radius. This line s divided as a cosine scale, and ismarked on .opposite sides with the corresponding angles. The scale AA on the left of the line, or the A scale, is taken as positive upward from the center c of the theoretical circle and scale BB on the right, or the B scale, is positive downward from c.

scale are read from to 130 downward.

In the manner known to navigators, a Value of L is assumed andthe value of d is derivedr from the nautical almanac. For the present explanation, it will be assumed that both L and d are positive in sign, or of the same name. For anyY particular problem, the value of L-'-d' is set off on the B scale. From the point L-d a line is drawn of a length likewise equal to two radii of the-,unit circle making any angle qi with the AB scale. For convenience, :and because the most accurate results can be obtained in this manner, the yangle 0 is made equal to 60, but the particular angle used inno way affects the calculations. y v ,f f

This second line, which is marked HA and will be calledv the HA or hour angle scale, is now divided as a cosine scale in such a manner that' hand end of the HA scale to meet the scale Ay 55 at b at a value equal to L-i-d. From a'point 2In other words, the angle markings on the Av having thevalueA ton the hour anglescale, (taken as positive, that is, betweenj() and 90) draw another line parallel to ab intersecting the Vertical rscale at some distances@ downward from the center c. This produces simil ar triangles a. b.

L-d, and t, m, Lfd; `From the law of similar triangles, we may -then write the ratio:

Solvingthisequation fora: we obtain the equation f The vsecond member ofvthis Equationl (c) is identicalwith the second member of Equation (b) above vandnwe nd that if :cV is read onthe BY l Y and positive the l conditions of. liorn'iulaA (b.) will beffullled and the equation accurately` solved as to signsvand scale (a cosine scale) as cos e functions.

Now it is immaterial'whether we write,- (+r) or +(-.'L'); and cos t (positive) lor -1-cos t (negative).. For this reason the above chart isapplicable to allcases-where L and d have the samev sign.

vI L and comes:

In obtaining-this equation, LV-i-d must be laid oi on the B scaleand L-d on the yA sealer' Solving for-x, we obtain the following formula:

ricos-":sin Lf'sin d-i-cos Lcos d cosV t which is the proper :formula for this case.

Thisl equationA coverseverycase where L and@ have opposite signs.-

`According to the mulae, my device as explained above 'equations and Y for# below will solve Referring now to the appara tus itself for car-.-

rying outthe above procedure, it is shown' in the accompanying drawings-in which Figs; '1 to 3 are Vgrapl'is for explaining the mathematical Vtheory upon which my invention is based. I l

Fig. 4 isl a top plan view of my device.

ions on the lines d rhave opposite signs, Equation I bey Fig. 8 is an enlarged plan view of the point of connection between the scales.

According to the invention, the device comprises rst a pair of scale members each divided as cosine scales. One of these scale members ID represents the hour angle or HA scale of Fig. 1 and carries a scale I I divided as a cosine scale starting with at the point I2 and extending to 180 at the point I4. The 0 end of the scale member I0 isrprovided with an enlargement I6 havingtherein a groove I8 running at an` angle of approximately 60 to the long axis of the scale member I. On the side of the groove 0U- Y posite the scale is arranged a block 28 which is able block 20.

v0 at the opposite orVV free end.

the .AB scale, correspondstothe vertical or AB scale of Fig. 1. Along its upper edge it carries a series of graduations forming a cosine scale 32, while the body is marked with two sets 34and 36 of degree markings.V These Ymarkingsrun in opposite directions, the marks 34 running from 0 at the right end to190"lnthecenter andto extend from 0vin the center to 90 at each end.

' of the' B and A scales oir Fig. `l.

for the remaining scale will be explained below.

dal'

The right hand and the left hand parts lof scale 34 thus correspond to the equivalent sections The scale 30 `slides in'theslot I8 and can be adjusted so that point` on it'willjbe opposite the endV I2 of the hour angle scale II. `The set screw 28 may then be tightened, and the whole arrangement willrepresent the two scales shown" in Fig. 1.

Suppose now that the right part of scale 34 is` set until'the point I2 strikes itat a value 'of L-d, where L and d are of the same sign. This corresponds to the setting of the point L -d in Fig. 1.

At the opposite or free end ofV the HA seal l I0, and exactly on the point I4 representing the Y 180 mark of this scale, is provided a pivot on which turns a transparent member f whicli may be imoved'by means of a handle around its pivot.

42 Yupstanding from the scale member` III'Y and a `Vconical ring 44 fixed inthe plate 3,8. `This conical pivot makes it possible forthe device to retain its accuracy in spite of any wear, as an accurate centering of .thevplate 88 on its pivotis requisite `to the accuracy of the whole device, Y

The plate 38 carries a line which extends exactly` through the center ofV the pivot or in other words through; the end or 180 point of the HA scalel I I. The kother endV of plate 38 overlies scale member 30 and ,obviously line 46 may/'swing along this scale to strike various angle values on the delineations 32. The `inner edge, 48, of the member 38 vforms an angle with the line V46 for a purpose to be explained below, .the` member 38 preferably tapering from its pivotedg end I4 towards its other end.

If nowthe plate 38 is moved until the line 46 strikes on the left part of scale 34 (A scale) a value L-i-d, and is held in this. position by a Markings 36 The reason u The pivot at I4 is preferably aconicalr one, being in the nature of a vconical member suitable clip 50, it will be seen that the line 46 will correspond to the line a, b on Fig. l.

The nal element of my device is a plate 52 of transparent material having an edge 54 adapted to cooperate with edge 48 of plate 38 and having a series of spaced parallel lines 56 forming the same angle with edge 54 as does line 46 with edge 48, so that when edges 48 and-54 are placed to#` gether lines 56 will always be parallel to line 46. If now one of the lines 56 is set'on the HA scale at a Vvalue t, beingless than 90 andtherefore positive, it is evident that the particularline 56 so set on the value t will correspond to the line tr of Fig. 1.

In actual operation, it is only necessary to note which of the lines Y56 is nearest the value tin the problem to be solved. By sliding plate 52 in one "direction or the other, this line because of the angular edges 48 and 54 will be shifted transversely until it strikes valuert on scale II.

Now from Equation (b) if we set L-don the` B scale 34 opposite the point I2; and set line 46r to a value L-l-d on the A` scale 34 and set member 52`sothat oneof its'lines 56 passes through the value t on scale III,- it is evident that the same line 56 will strike on the B scale 34 the value :r or cosfe. In this manner, by a proper setting, the

Vvalue of z can be obtained directly where L, d and tare known.

Where L and d are of opposite sign,it is merely vnecessary to set the value L-i-dvon theB-scale 34 tothe point I2 and to set the line 46 at a value-L-cl on the A scale 34.Y The line 56 passingvthrough the value twill then give the valuel easy ,sliding of the scale in the slot VI8 to its de.-V

sired position of adjustment.

In order to render reading more accurate, I may provide opposed Vernier sections 62 and 64 on the edges 48 and 54 respectively, scale'62 for example having 49 spaces equal tof50 spaces on scale 64;;

This vernier is used for example by first taking the vernier reading at the particular setting which it is desired'- to'measure and thenv the readings at the two nearest scale divisions on opposite sides of `this point.V The Vernier readings must then be made'progressively greater from theV lower scale .division through the setting to be read and the higher scale division. In other words, if the readings at the lower'and higher scale divisions should be 41 and 34 respectively and in the' center at the point to be measured 14, the proportions would then be 4l, 14, 34'. These are not progressively greater, and in order to'makethem so must be added to each of the secondV and third members. The proportions will thenV be 4l', 64, 84. The proportion of the space between the two scale readingsV of the point to be measured is: i

This may be 10' interval is then added to the value 'of the lower marking. i

The device may7 also be used for solving for the azimuth and declination. Referring. to Fig. 2,?

we assume that Z isfthe azimuth from'the elevated pole'. as indicated; In order to 1lnd;Z,'thev triangle should, be rotated until Z occupies the? position of t.' The sides of the4 triangle .about Z are then e and 90 .-L. Both of these will be on thesam'e side 'of the equator since'they are both less than 90 for a visible body. By analogywith Equation (I) and referring to .the type of chart shownin Fig. 1,"we must set L-'h on the B scale and L-I-h on thev A scale.. The' correct'azimuth from the elevatedpole will then vbeshown on the l HA scale opposite (or, in -other words, on a' line parallelto the'abline) toJ90-don the B'scale.

In other words, if the line 46 is set to L-I-'h on the 1 A scale 34 and the B scale' 34 is set at L-hto the point I2, then one of the parallel lines 56 will be set to 90-d on the Blscale34 and thi's'same parallel line l'will strike the value of Z on the scale I I.

In the case shown in Fig.43 lwhere L and d are oi-` different Isign, the azimuth will be read opposite 90-I-d on the B scalel 34. L-h is 'still set on B and L-I-h on A.

In order to obtain the azimuth about the depressed pole, it isfnecessary to solve theIco-lunar triangle drawn about t. An analysis of the cases which can arise` by the rules for co-'lunar'triangles will show that in such asituation we must reversel the setting s'above, setting-L-I-h von B (or setting B scale 34 'at'L-I-h, on-point'IZ) and L-'71,l on A (or setting line 46 lat Lhvon A scale 34) The azimuth may then be read on vscale II opposite 90-I-d`on`B (or onl line 56 vset lto 90-I-d onB scale 34) if L and 'd are of the lsame sign and opposite 90-d on Bv scale 34 if the signs are contrary. I f

In the problems of navigation, very fewquestions normally arise of angles greater than 90, particularly as for as the A and B scales are concerned. For convenience, therefore,these scales L are not divided and set oil up to 180 in the same manner as in Fig.` l. Instead,r scale 34 forms both the A and B scales, each reading from to 90, the 'B scale starting with`0 at the righthand end andtheA scale with 0 'at the left hand'end of the scale member 3D. For a visible body, Ythe setting on B can never exceed 90, so that there isno need to extend'the 'scale beyond this point. Alongside'the scale B on 34is a second scale h or 36, the'graduations of whichare in the opposite direction to those of scale V34; On this scale, the

.altitudes can vbe read directly, instead of by subtraction from the zenith distances which are given by scale B. y

The value L-I-d which is read on the A scale may sometimes exceed 90 although such cases are rare. Alongside the A scale therefore I havearranged the other portion o-f scale 36, Aas a d scale, the graduations being in the opposite direction from those on A. If it is necessary to read a value of more than 90 on A, 90 may be subtracted from the amount to be read and the difference read on h. A value of greater than 90 on B can be likewise read by subtracting 90 and reading on d.

According to the invention, it is possible among other problems to nd the following values:

l. Altitude or hour angle.

2. Azimuth from either pole.

3. Declination from azimuth, used in star identification.

.4.Gfreat` circle distances and courses ;V and the.y

location of intermediate longitudes'andlatitudes..

5. Hour angle of sunset or sunrise.. 4 Y The device will not solve directly, the original triangles for very large'hour angles or azimuths, those greater than'l1'5, since the various parts of the device cover the graduations for these quantities on the scalel I. I-Iowever, ifgthe'original triangle itself cannot be solved, it is always possi-.

ble to solve one of Ythe `three ,co-lunar trianglesdrawn .about it by 'simply reversing the normal settings and using the supplements of the sidesv and angles.V

The ycurrent practice with morst navigators is to lay -off the line of position from an altitudedifference and an azimuthand our newer `tables are constructed with thisfmethod in View. When the vdevice is available, however, this method is not the simplest one, as it involves .a separate solution' for azimuth which is unnecessary sincer an equally correct line of position may be laid off',

in most 'cases from a single setting.l

In sights near the meridian we may determine two altitude-diierences from two assumed longitudes when the line of position is a tangent to the two small circles drawn with these altitude-differences as radii andthe assumed positions as centers'. I

. In many sights we may solve once for longitude from the observed altitude (time sight) givingfone point on the line of position. We thenf'assume another longitude` at the samelatitudeand get an altitude-difference from that position drawing the line of positionfrorny the rstpointftangent to the small circle with. the altitude-diferenceas Near the prime vertical-neither ofthe above4r methods will work; in such a case use the original Sumner method of assuming two latitudes and" solvingk for longitude from each. This involves a change in setting but as the change is slight, it is simpler than making a solution for azimuth.- 1

To'plot a circle of position ,by tabular computation is a laborious'process, but if so desired such a circle may be very readily laid out by the device assuming, say, 'three' different latitudes rather far apart and solving for longitude from keach, thewcirclejof; position'so determined being, peculiarly satisfactory forvhigh sights where the dead reckoning position was also much in doubt.v

vIn solving for azimuth, it is best to solve the rproblem from the pole nearest in azimuth as the device will always solve-this problem.

f Most problems involving the hourangle of the sun may be solved directly, but in the case of a circumpolar, star it is `sometimes necessary yto get the hour* angle from the depressed meridian (or 180-t). Furthermore, in laying outgreat circle distances it is sometimes necessary to get the distance from a point vertically opposite the position coming backward along the track.

While 4I' have described herein one embodiment of my invention, I wish it to be understood that I do not intend to be limited by the description except within the scope of the appended claims.

l. A device for solving spherical triangles or the like comprising a pair of scales arranged at an angle with one another, at least oney of said scales being adjustable longitudinally to vary the point of intersection of the scales, and mechanical means for delineating a line intersecting both of said scales and parallel to a line passing of yintersection between the scales, a member piv` oted atfthe end of the second scale and'intersectfing the vfirst scale, whereby said member may ini-- Ldiente-aline drawn from theend of the secondi scale to a variable point on thek irst scale, and

meansfto delineate a line parallel to said irstlineV and intersecting both said scales, said second line being shiftable along the scales so that a.` point may be set on one of the scales and a corresponding value read on the other scale. f

3. A geometrical instrument for solving spheri cal triangles and the like, comprising a scale, a second Vscale intersecting said irst scale, said rst scale. being movable longitudinally to vary the point of intersection with the second scale, ',an element pivotedfat the end of the secondscale opposite its point of intersection with the rst scale and having a line thereon which intersects the ilrstscale, said element having an edge oblique to said line, and a second element hav-V ing a plurality of parallel lines thereon and hav# ing an edge of the same Obliquity to its lines, said member being-movable across said scales whereby the'lines` thereon may be shifted butwill always be parallel to the line on said lilrst' member.

y4. Aidevice for solving spherical triangles Aor' VVthe like, comprising a iirst scale divided as a cosine scale and an hour angle scale divided as a cosine scale from 0 at its innerendto 180 at its free end, said iirst scale intersecting said.

second scale at its zeroV point and movablelongitudinally so as to adjust the point of intersection between the scales, atransparent member pivoted at'the free end of the `secondscale and having a line thereon Vpassing through said pivot andintersecting the first scale whereby the point of intersection `of the line with the tlrst scale may be varied, said member having its edge toward the point of intersection of the scales oblique to said line, a second member of transparent material havingV a plurality -of parallel lines` de lineated thereon intersecting bothsaid scales and having an edge of the same obliquity. with respect tothe lines thereon as'the Obliquity of the edge of the first member to the line onthe-flrst member, whereby said members may be .moved to shiftA the parallellines on the second member transversely thereto while maintaining said lines always'parallel to thev line on the first member.` y 5. InV a mathematicalV instrument, a pair; of y scales arranged at an angle with one another,

means to .represent a line passing through the end point of one of said scales andY intersectingv vthe other scale, and means to representa second line parallel to the first line and intersecting .both said scales. I i

6. In a mathematical instrument,` a. pair of scales arranged at an angle with one another,

means to Vrepresent a line passing through theendpointv of one of said scales and intersecting the otherscale, and means torepresent a second line parallel'to the rst lineand intersecting both said scales, the points of intersection of the first line with thesecond scale and of the` second 'linewithf both the scales being variable.

- `7.';In a mathematical instrument, Vapair ofl scales ,arranged at an angle with one` another,

Vmeans to represent aline passing through the end point of one ofsaid scalesand intersecting thel other scale, and means `to represent a secr' ond line parallel to` therst line and intersect-V ing both said scales, the points of intersection of the .two scales, of thejfrst line with the second scale 'and ofthe second line with both'the scales being variable. 8. In a mathematicalinstrumentya pairrof scales arranged` at anfangle Vwithone another,-V

and mechanical devices for representing continuouslyparallel lines movable towards and fromoneanother, one of which intersects the, Vilrstfoi said scales and passes'through aj fixedl point on the second scale and the other of whichlinesw intersects both v.the scales.

9. In a.mathematical instrument, apair of scales arranged at an angle with one another, one of said scalesvintersectingj thevfsecondrscale at axed point and ,being movableV longitudi-v` .nally tovary the intersection point on therst scale, and mechanical devices for representing` continuously parallel lines movable towards and from one another one of which intersects the first of said scales and passes through a fixed point on the second scale and the Pother of which lines,y

intersects both the scales. Y' y l0. A device for solving spherical triangles and the lilecomprising a pair of scales intersecting oneanother, a member pivotedVv at the end of the second scale and intersecting the first scale,

whereby said member may indicate aline drawn from the endgof the second scale to a variable point'on thev first scale, and means to delineate; i a line parallel to said rst line and intersecting both said scales, said second line being. shiftablealong `the scales so that av point may be setV on one of the scales and a correspondingvalue read on the other scale.

V. B. BUCK. 

